Reny Solution

For a compact game $G$, let $$ \Gamma=\{(s,v)\in S\times\mathbb R^I:v=g(s)\} $$ and let $\overline\Gamma$ be its closure. A pair $(s,v)\in\overline\Gamma$ is a Reny solution if, for every player $i$, $$ \sup_{t_i\in S_i} \underline g_i(t_i,s_{-i})\le v_i, $$ where $\underline g_i$ is the lower semicontinuous regularization of $g_i$ with respect to the opponents' variables.

References

• [MFoGT, Definitions 4.8.1 and 4.8.2] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Graph closure and Reny solution for discontinuous games.